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 JacobiP

 http://functions.wolfram.com/07.15.06.0032.01

 Input Form

 JacobiP[\[Nu], a, b, z] == (((-1)^(a + b + 2 \[Nu]) 2^(1 + a) Sin[Pi \[Nu]] Gamma[1 + a])/ (Pi (-b - \[Nu])! (1 + a + \[Nu]) Gamma[1 + a + b + \[Nu]])) (z - 1)^(-1 - a) HypergeometricPFQ[{1, 1, 1 + a}, {1 - b - \[Nu], 2 + a + \[Nu]}, 2/(1 - z)] + (Gamma[1 + a + b + 2 \[Nu]]/(2^\[Nu] (Gamma[\[Nu] + 1] Gamma[1 + a + b + \[Nu]]))) (z - 1)^\[Nu] Sum[((Pochhammer[-a - \[Nu], k] Pochhammer[-\[Nu], k])/ (k! Pochhammer[-a - b - 2 \[Nu], k])) (2/(1 - z))^k, {k, 0, a + b + 2 \[Nu]}] + (((-1)^(a + b + 2 \[Nu]) 2^(1 + a + b + \[Nu]) Sin[Pi \[Nu]] Gamma[a + \[Nu] + 1])/(Pi (-b - \[Nu] - 1)!)) (z - 1)^(-1 - a - b - \[Nu]) Sum[((Pochhammer[1 + b + \[Nu], k] Pochhammer[1 + a + b + \[Nu], k])/ (k! (1 + a + b + k + 2 \[Nu])!)) (Log[(z - 1)/2] + PolyGamma[1 + k] - PolyGamma[-b - k - \[Nu]] - PolyGamma[1 + a + b + k + \[Nu]] + PolyGamma[2 + a + b + k + 2 \[Nu]]) (2/(1 - z))^k, {k, 0, -b - \[Nu] - 1}] /; Element[1 + a + b + 2 \[Nu], Integers] && 1 + a + b + 2 \[Nu] >= 0 && Element[1 + a + \[Nu], Integers] && 1 + a + \[Nu] >= 0 && b + \[Nu] <= 0 && !IntervalMemberQ[Interval[{-1, 1}], z]

 Standard Form

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 MathML Form

 P ν ( a , b ) ( z ) ( - 1 ) a + b + 2 ν 2 a + 1 sin ( π ν ) Γ ( a + 1 ) π ( - b - ν ) ! ( a + ν + 1 ) Γ ( a + b + ν + 1 ) ( z - 1 ) - a - 1 3 F 2 ( 1 , 1 , a + 1 ; - b - ν + 1 , a + ν + 2 ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + ( - 1 ) a + b + 2 ν 2 a + b + ν + 1 sin ( π ν ) Γ ( a + ν + 1 ) π ( - b - ν - 1 ) ! ( z - 1 ) - a - b - ν - 1 k = 0 - b - ν - 1 ( b + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( a + b + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( a + b + k + 2 ν + 1 ) ! ( log ( z - 1 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( - b - k - ν ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b + k + ν + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( a + b + k + 2 ν + 2 ) ) ( 2 1 - z ) k + 2 - ν Γ ( a + b + 2 ν + 1 ) Γ ( ν + 1 ) Γ ( a + b + ν + 1 ) ( z - 1 ) ν k = 0 a + b + 2 ν ( - a - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( - a - b - 2 ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k /; a + b + 2 ν + 1 a + ν + 1 b + ν 0 z { - 1 , 1 } Condition JacobiP ν a b z -1 a b 2 ν 2 a 1 ν Gamma a 1 -1 b -1 ν a ν 1 Gamma a b ν 1 -1 z -1 -1 a -1 HypergeometricPFQ 1 1 a 1 -1 b -1 ν 1 a ν 2 2 1 -1 z -1 -1 a b 2 ν 2 a b ν 1 ν Gamma a ν 1 -1 b -1 ν -1 -1 z -1 -1 a -1 b -1 ν -1 k 0 -1 b -1 ν -1 Pochhammer b ν 1 k Pochhammer a b ν 1 k k a b k 2 ν 1 -1 z -1 2 -1 PolyGamma k 1 -1 PolyGamma -1 b -1 k -1 ν -1 PolyGamma a b k ν 1 PolyGamma a b k 2 ν 2 2 1 -1 z -1 k 2 -1 ν Gamma a b 2 ν 1 Gamma ν 1 Gamma a b ν 1 -1 z -1 ν k 0 a b 2 ν Pochhammer -1 a -1 ν k Pochhammer -1 ν k k Pochhammer -1 a -1 b -1 2 ν k -1 2 1 -1 z -1 k a b 2 ν 1 a ν 1 b ν 0 z -1 1 [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2001-10-29